Basics of spreading: Granovetter’s threshold model

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# Basics of spreading: Granovetter’s threshold model
### David Garcia, Petar Jerčić, Jana Lasser <br><br> <em>TU Graz</em>
### Computational Modelling of Social Systems

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<div class="my-footer"><span>David Garcia - Computational Modelling of Social Systems</span></div>

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# So far

**Block 1: Fundamentals of agent-based modelling**
  
- Basics of agent-based modelling: the micro-macro gap
  - Tutorial: ABM basics in Python with Mesa (session 1)
- Modelling segregation: Schelling's model
  - Tutorial: ABM basics in Python with Mesa (session 2)
- Modelling cultures
  - Exercise 1: Schelling's model and Pandas (session 1)

**Block 2: Opinion dynamics**
- **Today:** Basics of spreading: Granovetter's threshold model
 - Exercise 1: Schelling's model and Pandas (session 2)
- Opinion dynamics
  - Exercise 2: Threshold models (session 1) **Deadline: 20.04.2022**

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# Overview

## 1. Collective behavior

## 2. Granovetter's threshold model

## 3. Modelling online collective emotions

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# Collective behavior

## *1. Collective behavior*

## 2. Granovetter's threshold model

## 3. Modelling online collective emotions

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# More is different

<img src="Figures/MoreIsDifferent.png" width="750" style="display: block; margin: auto;" />
[More is different: broken symmetry and the nature of the hierarchical structure of science. Philip Anderson, Science (1972)](https://cse-robotics.engr.tamu.edu/dshell/cs689/papers/anderson72more_is_different.pdf)

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## Complexity Science: Complicated versus complex

.pull-left[
<img src="Figures/Cogs.jpg" width="750" style="display: block; margin: auto;" />
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.pull-right[
<img src="Figures/network-ge2bd7754a_1920.png" width="750" style="display: block; margin: auto;" />
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- A **complicated system** has many pieces with specific functions and well-defined relationships. It has been carefully **engineered or designed**.  
- A **complex system** is composed of many particles that interact following some forces or dynamics. Its behavior follows from **natural principles**.

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# Complexity in pop culture
.pull-left[
<img src="Figures/JurassicPark.png" width="290" style="display: block; margin: auto;" />
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.pull-right[
<img src="Figures/Matrix.jpg" width="310" style="display: block; margin: auto;" />
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# Collective behavior in complex systems

<img src="Figures/animals.jpg" width="600" style="display: block; margin: auto;" />
[Challenges and solutions for studying collective animal behaviour in the wild. Lacey Hughey et al. Philosophical Transactions of the Royal Society B (2018)](https://royalsocietypublishing.org/doi/10.1098/rstb.2017.0005)
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# Collective behavior in social systems

**Interaction-induced collective behavior:** when the strength of interaction between individuals generates macro behavior
  - Schelling's model: low tolerance triggers moves that lead to segregation
  - Alxelrod's model: cultural exchange leads to larger cultures or supports coexistence of few cultures

**Diversity-induced collective behavior:** when the differences between individuals create the environment for macro behavior to emerge
  - Today's case: interaction strength and average agent stays the same, only variance between agents is the driver
  - More on network models: different positions in a social network are a way to induce cascades

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# Granovetter's threshold model

## 1. Collective behavior

## *2. Granovetter's threshold model*

## 3. Modelling online collective emotions

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# Binary decisions and collective behavior

.pull-left[The riot toy example:
- A group of individuals is part of a demonstration
- Individuals have a threshold of how many others have to be rioting to join the riot
- If enough people are in the riot, individuals with lower threshold join too]
.pull-right[
<img src="Figures/Riot.png" width="500" style="display: block; margin: auto;" />
]

- Proto-opinion: just participate / not participate
- Other examples with binary decisions depending on size: Diffusion of innovations, rumors, strikes, voting, leaving a party, migration

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# Diversity in collective behavior

- How does the distribution of preferences (thresholds) in a population affect its collective behavior? 
- Knowing the preferences does not directly tell you how the population will behave, you need to analyze how the population behaves
- Aim: understanding groups beyond the representative "mean" member

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# Rational agents in collective action

Assumption: the decision to join the collective action depends on:

- Risk or cost of participating.
- Examples of risks and costs:
  - Risk of being jailed in riot
  - Wage loss in strike
  - Cost of technology adoption

- The benefit (potential or sure) of the action taking place.
- Examples of benefits:
  - Political change after demonstration
  - Profit out of adopting innovation
  - Political party winning an election

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# An example of spreading

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# Net benefit and thresholds

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Net benefit =  benefit - costs

- Threshold to join: Net benefit is >0

- benefits increase and costs decrease with more people in the action (monotonic net benefit)

- weaker assumption: there is only one crossing of zero in the function of net benefit vs people in action

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.pull-right[
<img src="Figures/Benefit.png" width="800" style="display: block; margin: auto;" />
Example of net benefit function from Granovetter (1978)
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# Toy example of Granovetter's model

- 100 Agents
- Uniform sequence of thresholds with integer values `$[0,99]$`
- First agent activates, then second, and so on
- One agent joins per iteration and all agents are active in the end

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## Toy example of Granovetter's model version 2

- Same example as before but agents with thresholds `$1$` and `$3$` now have threshold `$2$`
- First agent activates and simulation ends
- Radically different outcome for minimal change in thresholds!
- Deducing preference distributions from collective outcomes is risky

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# Analyzing the distribution of thresholds
.pull-left[
- `$F(x)$` is the cummulative density function of thresholds:

$$F(x) = P(\Theta_i < x) $$
- `$r(t)$` is the number of active agents at time `$t$`
- `$r(0)$`: number of "instigators"
- Simulation reaches an equilibrium:
$$ r_e = F(r_e)$$
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.pull-right[
<img src="Figures/CDF.png" width="600" style="display: block; margin: auto;" />
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# `$r_e$` versus `$\sigma$`

.pull-left[
<img src="Figures/STD.png" width="500" style="display: block; margin: auto;" />
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- Assumption: Thresholds follow normal distribution with `$\mu$` and `$\sigma$`
- `$r_e$`: equilibrium number of active agents (simulation ended)
- `$\sigma$`: standard deviation of distribution of thresholds
- Number of agents is constant: 100
- `$\mu$` is constant: 25

- Sharp increase in `$r_e$` at a critical `$\sigma$` value: phase transition
- Diversity-induced collective behavior
]

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# Equlibrium in thresholds with `$\sigma=1$`

Low variance: thresholds concentrated around 25, no collective action
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# Equlibrium in thresholds with `$\sigma=10$`

Higher variance but equilibrium is still at low value

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# Equlibrium in thresholds with `$\sigma=12$`

Equilibrium starts to grow to small values
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# Equlibrium in thresholds with `$\sigma=13$`

Sharp change to upper equilibrium with very high value
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# Equlibrium in thresholds with `$\sigma=60$`

Slow decrease of equilibrium point towards very high variances
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## Granovetter's model: take home messages

- **Modelling action as rational choice:** thresholds as points where benefits outweigh costs or risks

- **Diversity matters:** Two populations with the same average threshold have very different behaviors even if mean threholds are the same

- **Tipping point or phase transition:** behavior changes dramatically at a narrow range of standard deviation of thresholds

- **Size effects:** small changes in threshold sequences can be important. When the population is small, you have a probability of very different outcomes. Inferring the preferences from the outcome is very hard and/or misleading

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# Modelling online collective emotions

## 1. Collective behavior

## 2. Granovetter's threshold model

## *3. Modelling online collective emotions*

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# Collective emotions

**Collective emotions:** Emotional states shared by a large amount of people at the same time

[Collective Emotions, Christian von Scheve and Mikko Salmela, Oxford University Press (2013)](https://global.oup.com/academic/product/collective-emotions-9780199659180?cc=at&lang=en&)
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# Collective emotions on social media

[Kony 2012 Timeline [Infographic], Chris Holden, The Huffington Post](https://www.huffpost.com/entry/kony-2012-timeline_b_1387729)

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## Quantifying emotions: valence and arousal

.pull-left[![](Figures/circumplex.png)]
.pull-right[

- **Valence:** the degree of pleasure experienced in an emotion 
  - Explains the most variance from positive/pleasant to negative/unpleasant

- **Arousal:** the level of activity associated with an emotion  
  - Explains less variance than valence but it is informative to differentiate emotions

]

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## The Cyberemotions modelling framework

- Horizontal: agent design
  - `$v$`, `$a$`: internal valence and arousal emotional state of the agent
  - `$s$`: visible emotional expression as measured (e.g. pos/neg/neu)
- Vertical: interaction between agents
  - `$h$` is a communication field averaging recent expression of agents
  - Agent's emotions change with time and the value of `$h$`
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# Valence and arousal dynamics

- `$b,d$`: baselines of valence and arousal
- `$\gamma_v$`, `$\gamma_a$`: relaxation tendency of valence and arousal towards baselines
- `$\xi_v$`, `$\xi_a$`: stochastic components of valence and arousal dynamics

[The dynamics of emotions in online interaction. David Garcia, Arvid Kappas, Dennis Küster and Frank Schweitzer. Royal Society Open Science (2016)](https://royalsocietypublishing.org/doi/10.1098/rsos.160059)

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# Examples of field influence functions

- Field influence functions as products of a polynomial of valence and arousal

- Approximation to some unknown function to fit empirically

- Valence depends on `$h$` (pos/neg) and arousal on `$|h|$` (absolute value)

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# Activation function

`$$\large s_i(t) = f_s(v_i(t)) \Theta [a_i(t) - \tau_i]$$`
- `$s_i(t)$` is the visible emotional expression of agent `$i$` aggregated in field `$h$`

- `$f_s(v)$` is a function of how valence is expressed through text
  - For example only sign or an approximation of internal valence

- `$\tau_i$` is agent `$i$`'s arousal threshold to expression

- `$\Theta[x]$` is the Heaviside step function: it has value 1 if `$x>0$`, and 0 otherwise

- After expression: reset of arousal to baseline ( `$a_i(t) \leftarrow d$` )
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# Behavior in simulations

[An agent-based model of collective emotions in online communities. Frank Schweitzer, David Garcia. The European Physical Journal B, 2010](http://www.springerlink.com/index/10.1140/epjb/e2010-00292-1)

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# Calibration experiment setup

- Study 1: reading pos/neg/neu threads and self-reports at home
- Study 2: reading pos/neg/neu threads and self-reports in the lab
- Study 3: reply to pos/neg/neu threads and self-reports before/after
- Self-reports include valence and arousal ratings and intention to participate in discussion and to continue reading the thread

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# Valence result

- Dynamics well fitted by a linear function with intercept shift depending on thread polarity ( `$h$` ). Natural decay of valence with `$\gamma_v = 0.37 \quad  [min^{-1}]$`
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# Arousal trigger results

- Dynamics well fitted by a linear function with intercept shift depending on thread abs value ( `$|h|$` ). Natural decay of arousal with `$\gamma_a = 0.41 \quad  [min^{-1}]$`
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# Empirical expression function

- Probability of post being classified as positive vs not positive and negative vs not negative in logistic regression (SentiStrength output with threshold)
- Function of valence but independent of arousal
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# Expression trigger and effects

- Probability to participate is flat up to a threshold of arousal, then it grows
- Effect of expression of arousal is a linear approach to baseline, not a resert

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# Summary

- Collective behavior in social systems
  - Complex versus complicated systems
  - Interaction versus diversity-induced collective behavior

- Granovetter's threshold model
  - Interaction in a well-mixed system given preferences or thresholds
  - Macro outcomes can vary a lot for small changes in threshold values
  - Variance in thresholds leads to aggregated activation

- Modelling online collective emotions
  - The Cyberemotions modelling framework 
  - Activation dynamics based on arousal thresholds
  - Calibrating an emotions model with experiments

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# Quiz

- Which of these is complex and which is complicated?
  - An airplane
  - The Internet
  - The Web
  - A deep neural network
- For a given `$\sigma$`, does `$\mu$` change the outcome in Granovetter's model?
- How can you find the fraction of active agents at `$t=0$` in Granovetter's model?
- In experiments, what does sentiment on text depend on: valence or arousal?

**Please give us feedback!** Answers are anonymous and private
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